The zero section of the universal semiabelian variety and the double ramification cycle

In this paper, we establish the existence of periodic orbits belonging to any [Formula: see text]-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact [Formula: see text]-form [Formula: see text] admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any non-trivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran–Polterovich–Salamon capacity.

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A Kähler Einstein structure on the tangent bundle of a space form

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201002009 ◽

2001 ◽

Vol 25 (3) ◽

pp. 183-195 ◽

Cited By ~ 10

Author(s):

Vasile Oproiu

Keyword(s):

Riemannian Manifold ◽

Riemannian Manifolds ◽

Tangent Bundle ◽

Negative Curvature ◽

Space Form ◽

Positive Curvature ◽

Holomorphic Sectional Curvature ◽

Zero Section ◽

Constant Negative Curvature ◽

Constant Positive Curvature

We obtain a Kähler Einstein structure on the tangent bundle of a Riemannian manifold of constant negative curvature. Moreover, the holomorphic sectional curvature of this Kähler Einstein structure is constant. Similar results are obtained for a tube around zero section in the tangent bundle, in the case of the Riemannian manifolds of constant positive curvature.

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Piunikhin-Salamon-Schwarz isomorphisms and spectral invariants for conormal bundle

Publications de l Institut Mathematique ◽

10.2298/pim1716017d ◽

2017 ◽

Vol 102 (116) ◽

pp. 17-47

Author(s):

Jovana Duretic

Keyword(s):

Morse Function ◽

Triangle Inequality ◽

Floer Homology ◽

Zero Section ◽

Spectral Invariants ◽

Conormal Bundle ◽

Morse Homology

We give a construction of the Piunikhin-Salamon-Schwarz isomorphism between the Morse homology and the Floer homology generated by Hamiltonian orbits starting at the zero section and ending at the conormal bundle. We also prove that this isomorphism is natural in the sense that it commutes with the isomorphisms between the Morse homology for different choices of the Morse function and the Floer homology for different choices of the Hamiltonian. We define a product on the Floer homology and prove triangle inequality for conormal spectral invariants with respect to this product.

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EXTENDING THE DOUBLE RAMIFICATION CYCLE BY RESOLVING THE ABEL-JACOBI MAP

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748019000252 ◽

2019 ◽

pp. 1-29 ◽

Cited By ~ 2

Author(s):

David Holmes

Keyword(s):

Moduli Space ◽

Smooth Curves ◽

Fundamental Class ◽

Stable Curves ◽

Double Ramification Cycle

Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel–Jacobi map. This breaks down over the boundary since the Abel–Jacobi map fails to extend. We construct a ‘universal’ resolution of the Abel–Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.

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Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers

Canadian Mathematical Bulletin ◽

10.4153/cmb-2014-031-6 ◽

2014 ◽

Vol 57 (4) ◽

pp. 749-764 ◽

Cited By ~ 1

Author(s):

Renzo Cavalieri ◽

Steffen Marcus

Keyword(s):

Moduli Space ◽

Moduli Space Of Curves ◽

Intersection Numbers ◽

Hurwitz Numbers ◽

Suggestive Evidence ◽

Wall Crossing ◽

Double Ramification Cycle ◽

Correction Terms

AbstractWe describe doubleHurwitz numbers as intersection numbers on the moduli space of curves Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers and the wall-crossing phenomenon in terms of a variation of correction terms to the ψ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera 0 and 1).

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On equivalence of two approaches in index theory

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is008001004jkt026 ◽

2008 ◽

Vol 2 (3) ◽

pp. 445-452

Author(s):

V. Manuilov

Keyword(s):

Commutative Algebra ◽

Compact Manifold ◽

Pseudodifferential Operators ◽

Index Theory ◽

Index Theorem ◽

Continuous Functions ◽

Zero Section ◽

Order Zero ◽

The One

AbstractThe algebra Ψ(M) of order zero pseudodifferential operators on a compact manifoldMdefines a well-knownC*-extension of the algebraC(S*M) of continuous functions on the cospherical bundleS*M⊂T*Mby the algebra К of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphismTfromC0(T*M) to К, which plays the role of a deformation for the commutative algebraC0(T*M). Similar constructions exist also for operators and symbols with coefficients in aC*-algebra. Recently we have shown that the image of the above extension under the Connes–Higson construction isTand that this extension can be reconstructed out ofT. That is why the classical approach to the index theory coincides with the one based on asymptotic homomorphisms. But the image of the above extension is defined only outside the zero section ofT*(M), so it may seem that the information encoded in the extension is not the same as that in the asymptotic homomorphism. We show that this is not the case.

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Equivariant Maps from Stiefel Bundles to Vector Bundles

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091515000541 ◽

2016 ◽

Vol 60 (1) ◽

pp. 231-250 ◽

Cited By ~ 1

Author(s):

Mahender Singh

Keyword(s):

Vector Bundle ◽

Lower Bound ◽

Vector Bundles ◽

Fibre Bundle ◽

Cohomological Dimension ◽

Stiefel Manifold ◽

Zero Section ◽

Zero Set ◽

Compact Lie Group ◽

Equivariant Maps

AbstractLet E → B be a fibre bundle and let Eʹ → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and Eʹ – 0, where 0 is the zero section of Eʹ → B. Let f : E → Eʹ be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 𝕊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions.

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Automorphism groups of rational elliptic surfaces with section and constant J-map

Open Mathematics ◽

10.2478/s11533-014-0446-6 ◽

2014 ◽

Vol 12 (12) ◽

Author(s):

Tolga Karayayla

Keyword(s):

Automorphism Groups ◽

Zero Section ◽

Ground Field ◽

Elliptic Surfaces ◽

Singular Fibers ◽

Weil Group ◽

Biholomorphic Maps ◽

Rational Elliptic Surfaces ◽

Fixed Section

AbstractIn this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ℂ. The automorphism group of such a surface β: B → ℙ1, denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) ⋊ Aut σ (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut σ (B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B. The Mordell-Weil group MW(B) is determined by the configuration of singular fibers on the elliptic surface B due to Oguiso and Shioda [9]. In this work, the subgroup Aut σ (B) is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.

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Nearly free curves and arrangements: a vector bundle point of view

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004119000318 ◽

2019 ◽

pp. 1-24 ◽

Cited By ~ 1

Author(s):

Simone Marchesi ◽

Jean Vallès

Keyword(s):

Single Point ◽

Point Of View ◽

Line Bundles ◽

Zero Section ◽

Combinatorial Invariant ◽

Counter Example ◽

Free Arrangement ◽

Finite Set ◽

Interesting Family ◽

Arrangement Of Lines

Abstract Over the past forty years many papers have studied logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. Terao conjectured thirty years ago that when a curve is a finite set of distinct lines (i.e. a line arrangement) its freeness depends solely on its combinatorics, but this has only been proved for sets of up to 12 lines. In looking for a counter-example to Terao’s conjecture, the nearly free curves introduced by Dimca and Sticlaru arise naturally. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non-zero section that vanishes on one single point, P say, called the jumping point, and that this characterises the bundle. We then give a precise description of the behaviour of P. Based on detailed examples we then show that the position of P relative to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics.

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